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Classification systems and their lattice

100%
Radeleczki S.
Discussiones Mathematicae - General Algebra and Applications
|
2002
|
tom 22
|
nr 2
167-181
EN
We define and study classification systems in an arbitrary CJ-generated complete lattice L. Introducing a partial order among the classification systems of L, we obtain a complete lattice denoted by Cls(L). By using the elements of the classification systems, another lattice is also constructed: the box lattice B(L) of L. We show that B(L) is an atomistic complete lattice, moreover Cls(L)=Cls(B(L)). If B(L) is a pseudocomplemented lattice, then every classification system of L is independent and Cls(L) is a partition lattice.
2
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On interval decomposition lattices

63%
Foldes S., Radeleczki S.
Discussiones Mathematicae - General Algebra and Applications
|
2004
|
tom 24
|
nr 1
95-114
EN
Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in an ordered set. They are defined abstractly as closed sets of a closure system on a set V, satisfying certain axioms. Decompositions are partitions of V whose blocks are intervals, and they form an algebraic semimodular lattice. Lattice-theoretical properties of decompositions are explored, and connections with particular types of intervals are established.
3
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Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

63%
Dzik W., Radeleczki S.
Bulletin of the Section of Logic
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2016
|
tom 45
|
nr 3/4
EN
We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.
4
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Commutative directoids with sectionally antitone bijections

51%
Chajda I., Kolařík M., Radeleczki S.
Discussiones Mathematicae - General Algebra and Applications
|
2008
|
tom 28
|
nr 1
77-89
EN
We study commutative directoids with a greatest element, which can be equipped with antitone bijections in every principal filter. These can be axiomatized as algebras with two binary operations satisfying four identities. A minimal subvariety of this variety is described.
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